1. Field
Aspects of embodiments of the present invention are directed toward applications of a peridynamic differential operator.
2. Description of Related Art
Derivatives are of fundamental importance in many branches of science. They describe the way a data point, entity, object, etc., interacts (e.g., rate of change) with its surrounding media. The relationships between derivatives of involved quantities provide the knowledge and understanding of physical phenomena and enable the construction of governing field equations.
Although differentiation is a powerful tool, it lacks certain properties. For instance, differentiation is a local quantity, and it does not change when the size of the surrounding media shrinks to zero. This is not consistent with reality. Today's experiments frequently pose the length scale requirement, especially at the micro- and nano-scales. In addition, the use of a derivative is pertinent to certain requirements on the regularity of a data set. On the other hand, critical conditions are mostly connected with a surge of irregularities, breakages, and discontinuities. Capturing the physics of these evolutions may require detailed treatment of derivatives specific to each problem.
Although the differentiation process is usually more direct than integration in analytical mathematics, the reverse is often true in computational mathematics, especially in the presence of a jump discontinuity or a singularity. Integration is a non-local process because it depends on the entire range of integration. However, differentiation is a local process and sensitive to abrupt changes. In real life, a physical phenomenon may be only described by a discrete set of data measured from experiments, and it is necessary to calculate the derivatives, to distinguish the noise, and to determine the discontinuities. Any small noise in the measurements may be magnified in the derivatives. Furthermore, many inverse problems arising from mathematical and physical equations may require accurate numerical differentiations.
Approximate solutions of the governing differential equations of many physical fields may require the evaluation of derivatives at discrete points in the domain. Their solutions may become challenging due to the presence of higher order derivatives, abrupt changes in behavior, nonlinearity, and multi-scale resolution arising from characteristic parameters. Currently available techniques for numerical differentiation may be classified as difference methods, interpolation methods, and regularization methods. These techniques may yield satisfactory predictions for smooth and precise variations without scatter. The regularization methods, for example, may express the derivative as the solution to the Volterra integral equation, and reduce the integral equation to a family of well-posed problems that depend on a regularization parameter. However, the determination of the optimal value for this parameter may not be a trivial task.
Image and data recovery may involve highly resource-intensive calculations to maintain accuracy, which may cause strain on a computer system and slow performance, particularly in situations where large sizes of images or data files, or large quantities of image or data files, must be recovered. This may cause inconvenience to users and may potentially damage computer processors by causing issues such as overheating during large batch operations. As such, accurate approximation methods and systems for use with computationally intensive tasks such as image and data recovery may be desired.